Combination Difference Synchronization between Identical Generalised Lotka-Volterra Chaotic Systems

This manuscript investigates the combination difference synchronization between identical Generalised Lotka-Volterra Chaotic Systems. Numerical Simulations have been performed which are in complete agreement of theoretical results.


Introduction
CHAOS is one of the significant features of nonlinear phenomenon. Chaos has been described in many ways, some describe it as a disorder appearing in the events so that they appear unpredictable. Chaotic behavior exists widely in engineering, biology, economics and many other scientific disciplines. As initially chaos was not known to be so important in any field but it was only after the ice-breaking work of Pecora and Caroll who gave the concept of synchronization and synchronized two chaotic systems by designing suitable controllers. With time many synchronization schemes were developed and currently many new schemes are being developed also. Some familiar schemes used are anti-synchronization, compound synchronization, complete synchronization, combination synchronization etc. Many techniques are used to achieve the above mentioned type of synchronization like adaptive control method, active control method, tracking control method etc. [1][2][3][4][5].
In this article the combination difference synchronization between identical chaotic systems has been achieved. The theoretical results are verified graphically which clearly exhibit that the technique used is effective and reliable for synchronizing the considered systems. We have arranged the remaining article as: Sec 2: formulates the problem; Sec 3: develops the synchronization theory. Sec 4: conducts the combination difference synchronization scheme [6][7][8]. Sec 5: consists of the discussions regarding numerical simulations and displays the results performed in MATLAB. Sec 6: concludes the article.

Problem Formulation
The scheme of combination difference synchronization requires two chaotic drive systems and one response system. Let the one drive system be ̇= ( ) (1) and the other drive systems be ̇= ( )

Synchronization Theory
We now develop the theory for combination difference synchronization among two chaotic drive systems (1)-(2) and one chaotic response system (3). We design the control function as We choose ( , , … , ) in such a way such that ( ) is negative definite. Therefore, by Lyapunov Stability Theory, we get lim ‖ ‖ = 0. Hence, the master systems and slave system are now synchronized.

Conclusion
In this paper, combination difference synchronization has been performed on identical chaotic Generalized Lotka-Volterra Systems. This type of synchronization technique can be used to determine the effect of some coexisting species on some particular species represented by slave system of Generalized Lotka-Volterra model. Further, the theoretical results are in excellent agreement with computational results.